How do you identity if the equation #9x^2+4y^2-36=0# is a parabola, circle, ellipse, or hyperbola and how do you graph it?

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Answer 1 · 2017-01-09T16:47:40

This is the equation of an ellipse.
See the graph below

We can do the discriminant test

#Ax^2+Bxy+Cy^2+Dx+Ey+F=0#

Discriminant, #Delta=B^2-4AC#

Here,

#9x^2+4y^2-36=0#

#Delta=0-4*9*4=-144#

As #Delta<0#, we conclude that the equation represents an ellipse.

Let's rearrange the equation

#9x^2+4y^2=36#

Divide throughly by #36#

#(9x^2/36)+(4y^2/36)=36/36#

#x^2/4+y^2/9=1#

We can compare this to

#x^2/a^2+y^2/b^2=1#

#a=2# and #b=3#

The vertices are #(2,0)#, #(-2,0)#, #(0,3)#. and #(0,-3)#

graph{(x^2/4)+(y^2/9)=1 [-8.89, 8.89, -4.444, 4.44]}